math and the minds eye activities 4

Fractions on A Line Line segments are divided into equal parts as a means of introducing the divi-sion model for fractions.. Using an overhead transparency made from Activity Sheet A,

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Unit IV I Math and the Mind's Eye Activities

Modeling Rationals

Egg Carton Fractions

Egg carron diagrams are used as visual models to inrroduce fractions and

frac-tion equivalence

Fractions on A Line

Line segments are divided into equal parts as a means of introducing the

divi-sion model for fractions

Fraction Bars

The Fraction Bar"-'> model for fractions is introduced and used to discuss

frac-tion equaliry and inequality

Addition and Subtraction with Fraction Bars

Franion bars are used to illustrate processes for adding and subtracting fraccions

Multiplication and Division with Fraction Bars

Fraction bJ.rs are used ro illustrate processes for multiplying and dividing

fractions

Introduction to Decimals

\'\lith the aid of base 10 mimber pieces, the concept of a decimal is imroduccd

and decimal norarion is discmsed

Decimal Addition and Subtraction

Base I 0 number pieces are used to develop processes fnr adding and

subtract-ing decimals

Decimal Length and Area

Thl' dimensions and areas of rectangles are found and the distinction between

linear measure and area measure is discussed

Decimal Multiplication and Division

Base 10 number piece rectangles arc used m tlnd the produn and quorienr of

decimals

Fraction Operations via Area: Addition and Subtraction

Fractions are reprl'senred by areas of rectangular regions, and fraction sums

and ､ｩｈＭｃｮＺｮ｣｣Ｎｾ found by finding rhe sums and differences of areas

Fraction Operations via Area: Multiplication

Two Fractions are multiplied by viewing them as rhe dimensions of a recrangle

and their product as rhe rectangle's area

Fraction Operations via Area: Division

The quotiem of two fracdons is f(mnd by constructing a rectangle for which

the area and one dimension are given

ath and the Mind's Eye materials are intended for use in grades 4-9 They are written so teachers can adapt them to fit student backgrounds and grade levels A single activity can be ex- tended over several days or used in part

A catalog of Math and the Mind's Eye materials and teaching supplies is available from The Math Learning Center,

PO Box 3226, Salem, OR 97302, 1 800 575-8130 or (503) 370-8130 Fax: (503) 370-7961

Learn more about The Math Learning Center at: www.mlc.pdx.edu

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Egg Carton Fractions

Actions

1 Show the students an egg carton cut in half as an example

of how some people buy one-half dozen eggs

Ask the students how they could cut the

carton differently to get one-half dozen

Comments

1 It is common to see egg cartons that have been cut in half in supermarkets Using an overhead transparency made from Activity Sheet A, discuss different ways to

cut an egg carton to get one-half dozen eggs Encourage volunteers to come forth

to draw their methods

Here are three ways:

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Actions

2 Distribute copies of the egg carton diagrams (Activity

Sheet A) Ask the students to devise ways to subdivide the

carton to show one-third of a dozen eggs Extend this

ques-tion to include two-thirds, three-fourths and four-sixths of a

dozen

Comments

2 You may wish to begin by doing one or two examples on the overhead To diagram the egg carton fraction '213, for example, subdivide the carton into three equal parts,

fill two of those parts with eggs and then write '213 under the diagram As the stu-dents draw their diagrams have them write the fraction symbol next to their drawing There are many ways to do each of these, but here are some typical responses:

frac-Discuss enough examples so that students can relate fractions to egg cartons

Math and the Mind's Eye

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3 Ask the students to find other egg carton fractions of a

dozen, recording each fraction and its diagram When the

students finish, display and discuss the results

3 Unit IV • Activity 1

3 This activity works well in small groups Here are the 32 egg carton frac-tions that students usually discover:

0/12,1/12,2/12, 11/12,12/12 0/6, 1/6, 216, 5/6, 6/6 0/4, 1/4, 214, 3/4, 4/4 0/3, 1/3, 213, 3/3 0/2, 1/2, 212 Sometimes students want to add 0/1 and 1/1

to the list of egg carton fractions 0/1 and 1/1 are obtained by not subdividing the car-ton and filling all or none of it You may wish to add these two fractions to the list if

your students discover them

The zeroes, like 0/4, come from dividing an egg carton into 4 equal parts but not filling any of them

Because the question is open-ended there may be some ingenious answers Some students have obtained up to 90 egg carton fractions For example, they get elevenths

by removing one egg from the carton and then dividing the carton into eleven parts

Of course, they are no longer getting fractions of a dozen These interesting approaches should be acknowledged Seldom do students ask if 6/12 of a dozen is equal to 2/4 of a dozen As symbols they are different, but they do represent the same number of eggs The fact that some frac-tions are equivalent to others will be

addressed in Action 5

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Actions

4 Using the overhead transparency of the egg cartons, show

the students the following diagram and ask them to

deter-mine what fraction of a dozen eggs is in the carton

Have them determine the fraction of a dozen

in each of the following (you may wish to add

more examples) Discuss each case

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6 (Optional- Egg Carton Fraction Wall Chart) Print

each of the 32 egg carton fractions on an index card Make

13 large egg carton diagrams with a different number of

eggs in each Have the students help you arrange a wall chart

similar to the following:

a Before students arrive in class, scramble

a few fraction cards to see if students can spot and correct the errors

b Take a few fraction cards off the wall and select students to correctly replace them

In both warm-ups the students can be asked

to "prove" the correct replacement by ing an egg carton diagram corresponding to their fraction

sketch-Math and the Mind's Eye

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Actions

5 Use a diagram to illustrate that 1/2 and 2/4, of a dozen,

are both egg carton fractions that represent 6 eggs Ask

them what other egg carton fractions represent 6 eggs?

1f2

2f4

Ask them to separate the remaining egg carton fractions into

groups, so that each fraction in the same group represents the

same number of eggs You may wish to point out that

frac-tions in the same groups are called equivalent fractions

12112 6/6 4/4 3/3 212

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7 (Optional - Adding Egg Carton Fractions) Put this

diagram on the overhead and go through steps like the

follow-rng:

a Shuffle a deck of 32 fraction cards like the ones made for

the wall chart in Action 6

b Select one card and ask for a volunteer to sketch that

frac-tion of a dozen in the first egg carton and write the fracfrac-tion

below the egg carton (suppose it is 1/3)

1f3

c Select a second card and repeat the previous directions for

the second egg carton (suppose it is 2/4)

d Ask a volunteer to combine the eggs into one carton and

write a number for the fraction of a dozen eggs that result

lli!E+-=m= fllllfiJ [[[[[8

Give each student a copy of activity sheet B and then repeat

this activity by drawing cards and having students record the

fractions and adding individually

7 (See below.)

Two egg cartons have been placed to the right of the equal sign on Activity Sheet B This is because it is possible that some frac-tion addition exercises generated by this activity will produce a full dozen and a frac-tion of a dozen as in this example:

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Egg Carton Recording Paper

EIHJI] EIHJI] EIHJI] EIHJI] EIHJI] EIHJI] EIHJI] EIHJI] EIHJI] EIHJI] EIHJI] EIHJI] EIHJI] EIHJI] EIHJI] EIHJI] EIHJI] EIHJI] EIHJI] EIHJI] EIHJI] EIHJI] EIHJI] EIHJI] EIHJI] EIHJI] EIHJI]

Activity Sheet /V-1-A Math and the Mind's Eye

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EHIIIJ+EHIIIJ = EHIIIJ EHIIIJ EHIIIJ+EHIIIJ = EHIIIJ EHIIIJ EHIIIJ+EHIIIJ = EHIIIJ EHIIIJ EHIIIJ+EHIIIJ = EHIIIJ EHIIIJ EHIIIJ+EHIIIJ = EHIIIJ EHIIIJ EHIIIJ+EHIIIJ = EHIIIJ EHIIIJ EHIIIJ+EHIIIJ = EHIIIJ EHIIIJ EHIIIJ+EHIIIJ = EHIIIJ EHIIIJ EHIIIJ+EHIIIJ = EHIIIJ EHIIIJ EHIIIJ+EHIIIJ = EHIIIJ EHIIIJ

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Unit IV • Activity 2

Fractions on a Line

0 v E R v

Actions

Part I Dividing Line Segments

into Equal Parts

E

1 Demonstrate the parallel line method of dividing a

seg-ment into equal parts

Figure 1

2 Distribute Activity Sheet A and a parallel line sheet to

each student Have the students subdivide the segments in

A Parallel-line sheet and Activity Sheets

A and B for each student An overhead transparency of each of these sheets

Comments

1 Place a transparency of equally spaced parallel lines beneath another transparency which has a line segment on it A master for a parallel line sheet is attached

Figure 1 shows the segment divided into 2 equal parts Figure 2 shows the segment divided into 3 equal parts

Figure 2

2 This can be done by placing a parallel line sheet with black lines under their acti-vity sheet and proceeding as you did on the overhead projector

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3 Draw the following diagram on the overhead

Tell the class you want to mark points to the right of S so that

each space (or interval) between points is the same length as

interval RS Ask how this can be done using equally spaced

parallel lines

, 3 Students may enjoy demonstrating their methods on the overhead Some may use 1 space for each interval

Others may use 2 spaces for each interval

4 Ask the students to use their parallel lines to solve there- 4 There may be more than one way to

maining problems on the Activity Sheet A Have the class dis- solve a problem

cuss their results

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Actions

Part II The Division Model for Fractions

5 Draw a segment representing 6 units on the overhead

pro-jector Use parallel lines to divide it into 3 equal parts Ask

the students how long each part is Tell them that we can also

speak of the length as six-thirds, 6/3

6 Duplicate the above 6 unit line and divide it into 5 equal

parts Ask the students to determine the length of each equal

part

7 Have students work problems 1 and 2 on Activity Sheet B

8 Put the following sketch on your overhead Ask the

stu-dents: IT AB is 2/3 units, how long is AC? Discuss

6 units

7 You may wish to do the first part of each problem on the overhead with class dis-cussion

8 Since 2j3 is the length obtained when 2 units are divided into 3 equal parts, then three lengths of 2j3 must be 2 units You may wish to include a few more examples here:

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9 Put this drawing on the overhead projector and ask the

students to help you

(a) determine the length of segment EG

(b) locate a point F so that EF is 1 unit long

4 Unit IV· Activity 2

9 EG must be 3 units long Because 3f7 results from dividing 3 into 7 equal parts, 7 lengths of 3f7 must be 3 units

To fmd point F, EG must be divided into 3 equal parts

10 Again, there may be more than one way to solve a problem

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Name _ Fractions on a Line

1 Use the parallel line sheet to divide each segment into the indicated number of parts

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1 Use the parallel line sheet to divide each segment into the indicated number of parts Then write a fraction name for each part

Length of One Part

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2 Show the students a sixth bar (red) with 4 parts shaded

Tell the students this is one model for "four-sixths" Write

the fraction 4j6 on the board and discuss the meaning of the

"top number" and the "bottom number"

3 Show the students, or have them select, several different

bars Have them describe each bar and then give its fraction

Fraction Bars® materials

are copy-righted products

of Scott Resources, Inc

of Fort Collins, Colorado

red, twelfths- orange) helps in ating bars with different numbers of parts

differenti-2 The top number, or numerator, tells the number of shaded parts The bottom num-ber, or denominator, tells the number of equal parts in the bar You may wish to defer the introduction of the terms "num-erator" and "denominator" until students are more familiar with fractions

3 Some students may describe a bar by giving its color and the number of shaded parts Others may tell the total number of

parts and the number of shaded parts

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4 Show the students a bar with all parts shaded, and have

them write the fraction for the bar Do the same for a bar

with no parts shaded Note that the fraction for the former

equals 1 and for the latter equals 0

5 Write a few fractions on the board (or overhead) and ask

students to visualize and then describe the bars for these

frac-tions For example, the bar for 3/8 has 8 equal parts and 3 of

them are shaded

6 Place a 3/12 bar under a lj4 bar Describe what this

shows Ask the students for other pairs of bars which have

the same amount of shading If the students have fraction

bars, have them sort the bars into piles according to their

shaded amounts

2 Unit IV• Activity 3

4 A fraction bar with all parts shaded is called a whole bar, and one with no shaded parts is called a zero bar

s;s = 1

Of4= 0

5 You may want to use fractions with atively small denominators, say, less than

rel-20 However, it may be instructive to see

if the students can describe bars for fractions such as 3fso, 99fwo, or lftooo

6 One part out of four has the same ing as three parts out of twelve If two bars have the same amount of shading, we say their fractions are equal Thus lj4 = 3ft2

shad-Math and the Mind's Eye

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7 Show the students a 1j4 bar and ask them to describe bars

with more parts but the same amount of shading Repeat this

activity for bars for 1/2, 3/4, 2J3, or other bars of your

choice Discuss with students the methods they use

8 Ask the students to sketch a bar for 2Js Then ask them to

divide the parts of this bar to show that 2Js = 6/15 Discuss

the relationship between numerators and denominators of

these two fractions

3 Unit IV• Activity 3

7 Some other bars that would have the same amount of shading as a 1f4 bar are bars for 2fs, 3/12, and 4/16 One method of forming these bars is to divide each part of

a 1f4 bar into an equal number of parts For example, dividing each part of a 1f4 bar into two equal parts doubles both the number of parts and the number of shaded parts on a

1f4 bar The result is a bar for 2fs

If transparency fractions bars are being used,

a transparency sheet can be placed over a bar and dotted lines for dividing the parts of the bar can be drawn on the sheet

8 Since each part is divided into 3 equal parts, both the number of shaded parts (the numerator) and the total number of parts (the denominator) are increased by a factor

of 3 The "new" numerator is 3 times the

"old" numerator and the "new" denominator

is 3 times the "old" denominator

2Js = s;1s

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Actions

9 Discuss a general method for forming equal fractions

Give several examples

10 Write the fractions 1/3 and 1/4 on the board (or

over-head) and ask the students to determine which is the greater

fraction Discuss their reasons Show the students a

sym-bolic way of writing their conclusion

11 Ask the students to determine the greater fraction for

each of the following pairs: 1f2 and 4/6, 5/12 and 2/3, 5f6 and

7/12, 2/3 and 3/4 Discuss with them how they arrived at

tal number of parts by the same factor, i.e multiplying the numerator and the denomi-nator of a fraction by the same· whole num-ber produces an equal fraction

10 One explanation that 1f3 is greater than

1f4 is that the shaded portion of a 1f3 bar is

larger than the shaded potion of a 1f4 bar Another explanation is that dividing a bar into 3 equal parts produces larger parts than dividing a bar into 4 equal parts

The statement "1/3 is greater than 1f4" can

be written as "1/3 > 1f4" This is equivalent

to the statement "1/4 is less than 1f3",

which is written "1/4 < 1j3" The inequality symbols,< and>, can be thought of as arrows which always point to the smaller number

11 If students are working with fraction bars, many of them will compare the shaded areas of the appropriate bars to reach their conclusions Others may arrive at conclu-sions without physically comparing bars For example, 4f6 > 1f2 because a 4j6 bar is more than half shaded; 5/12 < 2f3 because a

5/12 bar is less than half shaded and a 2f3 bar

is more than half shaded; 7/12 < 5f6 because

5f6 = 10/12 and 7/12 < 10/12; 2j3 < 3f4

because 213 = 8f12 and 3f4 = 9/12 Encourage students to fmd a variety of reasons to support their conclusions

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Fraction Bars for Thirds and Fourths

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I I I I I I I I I I I I I

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Addition and Subtraction

with Fraction Bars

Actions

1 Distribute fraction bars to each student or group of

stu-dents Write the following sums on the chalkboard or

over-head:

Ask the students to devise ways to use fraction bars to fmd

these sums Discuss the methods the students use

The students may use a variety of methods

to find the sums You may want to ask some of them to demonstrate their methods (a) One way to find this sum is to find a bar whose shaded amount is the total of the shaded amounts on the 1/4 and 213 bars:

1J4 + 2f3 = 11f12 Some students may replace the bars for 1/4

and 213 by equivalent bars which have the same number of parts:

1f4 + 2f3 = 3f12 + 8f12 = 11f12 This illustrates the process of adding frac-tions by expressing them as fractions with

a common denominator

Continued next page

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2 Ask the students to use their fraction bars to fmd

2 The subtraction can be done by fmding a bar whose shaded amount is the difference

of the shaded amounts on the 3/4 and 213

bars:

3f4-2f3 = 1f12

Another approach is to replace the 3/4 and

213 bars by equivalent bars with the same number of parts, and then fmd the differ-ence:

3f4-2f3 = 9f12- 8f12 = 1f12

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Actions

3 Have the students use the methods discussed to find

addi-tional fraction bar sums and differences

4 (Optional.) Ask the students to draw a sketch of a 4/9 bar

Then ask them to explore ways of using fraction bar sketches

to find lf3 + 4f9 Discuss

Comments

· 3 Here are some possibilities:

4 One way to obtain a sketch of a 4/9 bar

is to trace around a fraction bar to obtain a blank bar This blank bar can be divided into 9 equal parts and then 4 of these parts can be shaded You may want to discuss with the students how the blank bar can be divided into 9 roughly equal parts by sight

A fairly accurate subdivision can be tained by frrst dividing the bar into thirds and then dividing each third into thirds The students should recognize that sketches are an aid to thinking Even though their sketch of a 4/9 bar may not have 9 pre-cisely equal parts, they may use their sketches to help them visualize an ideal bar Two sketches for fmding 113 + 4/9 are shown here In the second sketch, the dotted lines indicate that a 113 bar has been converted to a 3/9 bar

ob-1f3 + 4fg = 7fg

1f3 + 4fg = 3fg + 4fg = 7jg

You may want to ask some of the students

to show their sketches on the overhead

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5 (Optional.) Ask the students to sketch a lfs bar and a lf2

bar Then ask them to subdivide these bars so they have the

same number of parts, and provide the names of the resulting

5 Dividing each part of the lts bar into 2

equal parts and each part of the lf2 bar into

5 equal parts results in tenth bars:

1fs = 2110

Subdividing bars to obtain bars with the same denominator is useful in the next action You may want to discuss with the students how other pairs of bars can be sub-divided to obtain the same number of parts, e.g a 1/3 bar and a 1/5 bar, a 1/6 bar and a

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Multiplication and Division

with CFraction Bars

frac-Comments

1 If fraction bars are not available, this activity can be done as a class discussion using fraction bar transparencies on the overhead

2 You may want to begin the discussion

by recalling the "repeated addition" model

of multiplication for whole numbers, e g

3 x 8 may be thought of as 8 + 8 + 8 or, what is the same, combining 3 groups of 8:

"one-fourth of eight."

(i) • • •

ｾ

1/4 x 8 = 1/4 of a group of 8 = 2

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8/12 bar can be thought of as a group of 8 twelfths Thus l/4 x 8/12 is l/4 of a group

Similarly, 3/4 x 8/12 can be thought of as 3/4 of a group of 8 twelfths This can be obtained by dividing 8 twelfths into four equal parts and taking 3 of them The result

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· 3 (a) Some students may report the answer

is 2/6, others l/3 Either is appropriate (b)

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4 (Optional) Have the students imagine or sketch fraction

bars to help them find the following:

is the same as the amount shaded on a 1/5

bar Other students may draw a sketch (b) Here are two possible sketches In the second sketch, a 3/5 bar has been converted into a 6/10 bar

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4 (Continued)

(d) One way to determine 2/3 of 4/5 is to first divide each part of a 4/5 bar into 3 equal parts This converts the bar to a fifteenths bar:

Then 2/3 of 4/5 can either be determined by ( 1) taking 2/3 of each of the original shaded parts:

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5 Ask the students for their ideas about using fraction bars

to find 8/12 + 1/6 Discuss, then ask for ideas to find

5/6 + 1/3

5 Some students may say the shaded area

of the 1/6 bar fits into the shaded area of the 8/12 bar 4 times Thus, 8/12 + 1/6 = 4

8 +2 = 4

Similarly, there are 2 2/3 groups of 3 in 8 {In the illustration below, the grouping on the right contains 2 parts of the 3 needed for a group Hence it is 213 of a group.)

ＸＸｾ

8+3=2213

Since there are 2 1/2 of the shaded regions

of a 1/3 bar in the shaded region of a 5/6 bar, 5/6 + 1/3 = 2 1/2:

5/6 + 1/3 = 2 1/2

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6 Ask the students to use fraction bars to find:

11/12 + 1/3 = 2 3/4

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7 Discuss with the students how 3/4 + 1!3 may be found by

replacing the 3/4 and 1/3 bars with bars which have a

com-mon number of parts Then ask the students to use this

method to find l/3 + 1!2 and 3/4 + 5/6

6 (Continued) (e) Only 2/3 ofthe shaded area of a 3/4 bar will fit into the shaded area of a 1(2 bar:

1/2 + 3/4 = 2/3 Note that in this statement, the quotient 2/3 does not represent the shaded area on a 2/3 bar Rather 2/3 indicates the portion of the shaded area of the 3/4 bar that fits into the shaded area of the l/2 bar

7 Comparing the 3/4 and 1/3 bars, it is apparent that the shaded area of the 1/3 bar fits into the shaded area of the 3/4 bar 2 and

a fraction times It may not be clear what this fraction is By replacing the bars by equivalent twelfths bars, one sees that this fraction is l/4

3/4 + 1/3 = 2?

3/4 + 1/3 = 9/12 + 4/12 = 2 1 /4 Dividing one fraction by another is fre-quently simplified by replacing the frac-tions by equivalent fractions with a common denominator

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Actions

Comments

of a 1{2 bar fits into the shaded area of a 1/

3 bar This can be seen by converting the bars into sixth bars

1/3 + 1/2 = 2/6 + 3/6 = 2/3

As seen below, the shaded area of a 5/6 bar

is greater than the shaded area of a 3/4 bar Hence, only a portion of the shaded area of

a 5/6 bar will fit into the shaded area of a 3/4 bar Thus 3/4 + 5/6 is less than 1

To fmd what portion of the shaded area of

a 5/6 bar fits into the shaded area of a 3/4 bar, the bars may be converted to twelfth bars This divides the shaded area of a 5/6 bar into 10 equal parts - 9 of these 10 parts equals the shaded area of a 3/4 bar Hence, 3/4 + 5/6 = 9/10

\._ _ ., _ _ )

v

9/10 of 5/6 3/4 + 5/6 = 9/12 + 1 0/12 = 9/10

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8 (Optional) Ask the students to use sketches of fraction

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Introduction to Decimals

0 v E R v E w

Actions

1 Distribute the base 10 number pieces to each student Tell

them that in this activity the largest piece represents the unit,

1 Ask them to determine the value of the other number

pieces Discuss

1

(unit)

2 Ask the students how they might devise base 10 number

pieces to represent 10 and 100

do the cutting Each student, or group of students, should have at least 11 unit, 15 tenth and 21 hundredth pieces

If the largest piece represents the unit, then the pieces have the following values:

1

1f10 (one tenth) (one hundredth)

2 One way would be to make a strip of 10 units to represent 10, and then join 10 of these strips side-by-side to represent 100

It may aid the students ization to construct number pieces for 10 and 100 and post them in view The hundreds piece will be a square meter an d

visual-the tens piece will be 10 centimeters by 1 meter

ｾ 10

(ten)

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